and 
.
The method used for the determination of was that suggested
by Hoyt et al [58]. The calculations are based on Eq. 
. This expression contains
a singularity at 
and an integral running from 
which is numerically
impossible to evaluate. However Hoyt et al showed that with certain approximations the singularity may be
conveniently dealt with and that by suitable choice of upper and lower integration limits a numerical solution can
be found.
Modifications were necessary
to the method of Hoyt et al due to the fact that the data in the present work was based on
fluorescence measurements whereas transmission EXAFS data were used for the original work. Secondly the criteria
established for the choice of the upper and lower energy integration limits were relevant to K-edge data and had to be
re-established for data around the 
edge.
Fluorescence data measured around the absorption edge of interest
from the crystal sample is first spline fitted and normalised such that the
resulting values are unity
far above the edge and zero far below the edge with the edge structure seen as fluctuations
about unity. Several possibilities exist for converting such experimental data into an
curve. The data may be fit directly to theoretical data for or it may first be fit to
atomic cross section data and in turn converted to using the Optical theorem. However
difficulties in measuring absorption thicknesses tend to introduce systematic error into atomic cross section
data [111]. For this reason the first method was used for fitting the experimental data in the region
of the absorption edge but since the available data did not extend to very high or very low energies
tables of atomic absorption data were used for the extrapolation to these energies.
In the vicinity of the absorption edge the normalised experimental fluorescence data
were converted into data by multiplying it into theoretical data
of Cromer & Libermann [25] calculated using CROSSEC. Since the experimental data only
extended out to about a hundred electron-Volts above and below the edge, a straight line fit to the theoretical
data was performed for the above and below edge regions to produce two linear representations of
, 
and 
as a function of energy. The
experimental curve was calculated by
Far above and far below the absorption edge the normalised data 
were converted into the
atomic cross-section, 
,
in the following way. Two forms of a 
- 
type equation for the mass
absorption coefficient above
and below the absorption edge were determined by inserting the coefficients 
,
given in the McMasters tables [85], into the following
Eq. [59] [85]
where 
is the atomic cross-section for energies above
the 
absorption edge (corresponding to 
, 
, 
, 
). The values of were summed at the
appropriate energies to yield the total cross-section .
Experimental values for were then derived from cross-sections using
the Optical theorem (see Sec. , Eq. )
The Kramers-Kronig relationship, Eq. , may be written in terms of energy as
The presence of the singularity at 
requires that the equation be manipulated
to allow integration numerically. Hoyt et al. split the integral into three parts
and deal with the singularity using a Taylor series expansion. The resulting numerically
calculable solution is,
The integration was carried out using Simpson's rule. The last term of Eq.
was summed only up to and including the 
term since higher terms were of the order of 
of the overall magnitude of the summation term. An additional correction term equal to 
, where
is the total energy of the atom in question, has been determined by Cromer & Libermann [25]
and is added to the resulting value. Values of for all atoms between 
and 
are tabulated in Table. II of [25].
The resulting program was initially tested with K-edge EXAFS data collected on an 
copper
foil. The data were collected
on the EMBL EXAFS beam-line in HASYLAB, DESY with an Si(111)
monochromator in the absence of any focussing mirror. The energy resolution of the beam line at the time was measured
to be 
. The data were placed on an absolute energy
scale [93]. To carry out a comparison with the copper K-edge data published by
Hoyt et al the same integration limits were used, i.e. an upper integration limit
of 
and a lower limit equal to 
.
The experimental curve of Hoyt et al and that obtained from the experimental
data of Pettifer were found to be very similar in
shape with only very minor differences in the fine structure. However, it was observed that the data
of Hoyt et al were on an arbitrary energy scale shifted with respect to calibrated data
by 
. This still allowed an idea to be obtained about the relative magnitudes of
the resultant data by applying an approximate energy correction to the data of
Hoyt et al.
Figure: Comparison of the results of the Kramers-Kronig transformation performed on data taken around the K-edge
of copper. The solid line represents the results of the present transformation program using data collected at the EMBL
EXAFS beam-line at an energy resolution of and placed on an absolute energy scale. The points are
taken from the
data published by Hoyt et al. Ignoring the mismatch in the energy axis there is discrepancy of
about 
in the value.
The results from both studies are shown in Fig. . The solid line in the result
of the present calculation and the set of data points shown are taken from the results of Hoyt et al.
The mismatch in energy of both sets data is clearly visible. When the data of Hoyt et al were shifted
along the energy axis and along the axis the best fit was obtained at values corresponding
to a energy shift of 
and 
. The agreement between the two sets of results is to better
than 
if the discrepancy along the energy axis is overlooked.
Having established that the results produced by the present program were in sufficient agreement with
those of the Hoyt et al, it was then necessary to re-establish the integration criteria for
use with edge data. A similar procedure as that used in the previous work was followed in that several
upper and lower limits of
integration were used and the point of convergence for the calculated data was sought.
The data used for this purpose were Pt absorption edge data collected on the EMBL EXAFS
beam-line from a sample of 
).
Figure: Resulting curves when the lower integration limit is held
at 
and upper limits of 
(top curve), 
(middle curve), 
, 
and 
edge energy are used. The lower curve is produced by the latter three of these demonstrating that
convergence can be achieved by using an upper integration limit of 
edge
energy.
Figures and show respectively the curves calculated
for various values of the upper and lower integration limits. Convergence of the calculated
values was achieved when using an upper integration limit of and a lower limit
of 
.
Figure: Results of the K-K transformation when the upper integration limit is kept
at edge energy and lower limits of 
, 
, 
, 
,
and 
edge energy (from bottom to top). The upper most curve arises from limits of
and thus demonstrating that a value of is
sufficient as a lower integration limit.
These integration limits were then used together on the cis-platin data and the resulting
curve was compared with the curve obtained by using the theoretical values for obtained
from the theory of Cromer & Libermann [25] implemented in the program CROSSEC.
The two curves are shown in Fig. . The deviation of the experimental curve
from theory is a result of the structure present in the XANES region of the absorption
spectrum.
Figure: Plot comparing values from the result of the Kramers-Kronig transformation
performed on experimental absorption data taken at the edge of platinum in cis-platin (solid line)
with those value calculated using the theory of Cromer & Libermann (dashed line).
